Mathematical aspects of the nuclear glory phenomenon: from backward focusing to Chebyshev polynomials

TL;DR

This paper explores the mathematical structure behind the nuclear glory phenomenon, revealing how Chebyshev polynomials describe the angular distribution of cumulative particles near the backward direction, leading to a characteristic focusing effect.

Contribution

It establishes a connection between the angular dependence in cumulative particle production and Chebyshev polynomials, providing a mathematical explanation for the backward focusing effect.

Findings

The differential cross section behaves as 1/√(π - θ) near θ = π.

Recurrent relations between polynomials are derived and linked to Chebyshev polynomials.

The nuclear glory phenomenon occurs for any multiplicity of interactions, elastic or inelastic.

Abstract

The angular dependence of the cumulative particles production off nuclei near the kinematical boundary for multistep process is defined by characteristic polynomials in angular variables $J_N^2(z_N^\theta)$, where $\theta$ is the polar angle defining the momentum of the final (cumulative) particle, $z_N^\theta = cos (\theta/N)$, the integer $N$ being the multiplicity of the process (the number of interactions). Physical argumentation, exploring the small phase space method, leads to the appearance of equations for these polynomials $J_N^2[cos(\pi/N)]=0$. The recurrent relations between polynomials with different $N$ are obtained, and their connection with known in mathematics Chebyshev polynomials of 2-d kind is established. As a result of this equality, differential cross section of the cumulative particle production has characteristic behaviour $d\sigma \sim 1/ \sqrt {\pi - \theta}$ at $\theta \sim \pi$ (the backward focusing effect). Such behaviour takes place for any multiplicity of the interaction, beginning with $n=3$, elastic or inelastic (with resonances excitations in intermediate states), and can be called the nuclear glory phenomenon, or 'Buddha's light' of cumulative particles.